A381053 Integers k such that Fibonacci(k) is odd and divides the sum of the first Fibonacci(k) nonzero Fibonacci numbers.

1, 2, 34, 46, 68, 92, 94, 106, 166, 188, 212, 214, 226, 274, 332, 334, 346, 394, 428, 452, 454, 466, 514, 526, 548, 586, 634, 646, 668, 692, 694, 706, 754, 766, 788, 886, 908, 932, 934, 1006, 1028, 1052, 1114, 1126, 1172, 1174, 1186, 1234, 1268, 1292, 1294, 1306

Offset: 1

Author

Oisín Flynn-Connolly, Apr 14 2025

Keywords

• nonn

Comments

Has infinitely many members.

Subsequence of A383021.

Contains all 2p and 4p such that p is an odd prime and p == 2,8 (mod 15).

Examples

			For k =2, Fibonacci(2) = 1, which is odd, and Fibonacci(Fibonacci(1)) = Fibonacci(1) = 1, which is divisible by 1.
For k = 34, Fibonacci(34) = 5702887 is odd, and Fibonacci(1) + Fibonacci(2) + ... + Fibonacci(5702887) = Fibonacci(5702889) - 1, which is divisible by Fibonacci(34) = 5702887.
		

Links

• Amirali Fatehizadeh and Daniel Yaqubi, Average of the Fibonacci numbers, J. Integer Seq. 25 (2022), no. 2, Art. 22.2.6, 10 pp.
• Oisín Flynn-Connolly, On the divisibility of sums of Fibonacci numbers, arXiv:2504.09938 [math.NT], 2025.
• Michal Křížek and Lawrence Somer, Period lengths modulo n and average of terms of second order linear recurrences, Integers 24 (2024), Paper No. A36, 41 pp.

Crossrefs

Cf. A000045, A158569, A331976, A383021, A331977.

Extensions

More terms from Alois P. Heinz, Apr 14 2025